Abstract
Disproportionality indices aim at measuring to what extent the composition of a parliament differs from the distribution of the votes among parties. Malapportionment indices measure to what extent the number of parliament seats attached to each district differs from the distribution of the population among districts. Since there exist many different such indices, some conditions have recently been proposed for assessing the merits of the various indices. In this paper, we propose a characterization of two disproportionality and malapportionment indices: the Duncan and Duncan index (also called Loosemore–Hanby) and the Lijphart index.
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Notes
In a recent paper, (Nurmi 2014) argues that the concept of proportional representation is both vague and ambiguous. To make things clear, we consider one-option balloting and proportional allocation of seats.
No disproportionality index has been characterized, yet several characterizations of allocation methods (i.e., techniques to allocate seats to parties in view of the electoral result) are available in the literature (Balinski and Young 1975).
We thus implicitly consider that the size of the set of voters does not matter.
The Duncan and Duncan index is often presented as \(1/2 \times \sum _{i \in N} \left| \pi _{i} - x_{i}/s(x) \right|\). The factor 1/2 is convenient in applications because it scales the index between 0 and 1. We purposefully drop this factor because it makes the index simpler. The difference between the two forms of the index is not more relevant than the difference between a length measurement in meters or in feet.
A characterization of the \(L^{1}\) distance (obviously linked to index \(f_{\text{ DD }}\)) already exists (Fields and Ok 1996). It is not completely relevant to our problem because the authors consider it as a distance between two points in \(\mathbb {R}^{n}\) while we are interested in the ‘distance’ between a point in \(\mathbb {N}^{N}\) and a point in \(\mathbb {Q}^{N}\) with the constraint that the sum of the coordinates be 1.
All proofs are deferred to Sect. 5.
Notice that \(f_{\text{ DD }}\), \(f_{\text{ DD }}^{2}\) and \(\exp (f_{\text{ DD }})\) all induce the same ranking. Actually, all strictly increasing functions of a given index induce the same ranking as that index.
The Rae index is defined as \(f_{\text{ DD }}/n\).
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Bouyssou, D., Marchant, T. & Pirlot, M. A characterization of two disproportionality and malapportionment indices: the Duncan and Duncan index and the Lijphart index. Ann Oper Res 284, 147–163 (2020). https://doi.org/10.1007/s10479-018-3073-y
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DOI: https://doi.org/10.1007/s10479-018-3073-y